Mathematics Grade 10 Teachers' Guide
Siyavula WebBooks
By:
Bridget Nash
Mathematics Grade 10 Teachers' Guide
Siyavula WebBooks
By:
Bridget Nash
Online:
< http://cnx.Org/content/colll341/l.l/ >
CONNEXIONS
Rice University, Houston, Texas
This selection and arrangement of content as a collection is copyrighted by Bridget Nash. It is licensed under the
Creative Commons Attribution 3.0 license (http://creativecommons.Org/licenses/by/3.0/).
Collection structure revised: August 10, 2011
PDF generated: August 10, 2011
For copyright and attribution information for the modules contained in this collection, see p. 47.
Table of Contents
1 Overview 1
2 Assessment and Assessment Support 9
3 Chapter Contexts 25
4 On the Web, Everyone can be a Scientist 29
5 FullMarks User Guide 33
6 Rich Media 37
7 Maths Blog Posts 43
8 Solutions 45
Index 46
Attributions 47
IV
Chapter 1
Overview 1
1.1 Overview
1.1.1 Curriculum Overview
Before 1994 there existed a number of education departments and subsequent curriculum according to the
segregation that was so evident during the apartheid years. As a result, the curriculum itself became one of
the political icons of freedom or suppression. Since then the government and political leaders have sought
to try and develop one curriculum that is aligned with our national agenda of democratic freedom and
equality for all, in fore-grounding the knowledge, skills and values our country believes our learners need to
acquire and apply, in order to participate meaningfully in society as citizens of a free country. The National
Curriculum Statement (NCS) of Grades R 12 (DoE, 2011) therefore serves the purposes of:
• equipping learners, irrespective of their socio-economic background, race, gender, physical ability or
intellectual ability, with the knowledge, skills and values necessary for self- fulfilment, and meaningful
participation in society as citizens of a free country;
• providing access to higher education;
• facilitating the transition of learners from education institutions to the workplace; and
• providing employers with a sufficient profile of a learner's competencies.
Although elevated to the status of political icon, the curriculum remains a tool that requires the skill
of an educator in interpreting and operationalising this tool within the classroom. The curriculum itself
cannot accomplish the purposes outlined above without the community of curriculum specialists, material
developers, educators and assessors contributing to and supporting the process, of the intended curriculum
becoming the implemented curriculum. A curriculum can succeed or fail, depending on its implementation,
despite its intended principles or potential on paper. It is therefore important that stakeholders of the
curriculum are familiar with and aligned to the following principles that the NCS is based on:
Principle
Implementation
Social Transformation
Redressing imbalances of the past. Providing equal
opportunities for all.
continued on next page
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1
CHAPTER 1. OVERVIEW
Active and Critical Learning
Encouraging an active and critical approach to
learning. Avoiding excessive rote and uncritical
learning of given truths.
High Knowledge and Skills
Learners achieve minimum standards of knowledge
and skills specified for each grade in each subject.
Progression
Content and context shows progression from simple
to complex.
Social and Environmental Justice and Human
Rights
These practices as defined in the Constitution are
infused into the teaching and learning of each of the
subjects.
Valuing Indigenous Knowledge Systems
Acknowledging the rich history and heritage of this
country.
Credibility, Quality and Efficiency
Providing an education that is globally comparable
in quality.
Table 1.1
This guide is intended to add value and insight to the existing National Curriculum for Grade 10 Math-
ematics, in line with its purposes and principles. It is hoped that this will assist you as the educator in
optimising the implementation of the intended curriculum.
1.1.2 Curriculum Requirements and Objectives
The main objectives of the curriculum relate to the learners that emerge from our educational system. While
educators are the most important stakeholders in the implementation of the intended curriculum, the quality
of learner coming through this curriculum will be evidence of the actual attained curriculum from what was
intended and then implemented.
These purposes and principles aim to produce learners that are able to:
• identify and solve problems and make decisions using critical and creative thinking;
• work effectively as individuals and with others as members of a team;
• organise and manage themselves and their activities responsibly and effectively;
• collect, analyse, organise and critically evaluate information;
• communicate effectively using visual, symbolic and/or language skills in various modes;
• use science and technology effectively and critically showing responsibility towards the environment
and the health of others; and
• demonstrate an understanding of the world as a set of related systems by recognising that problem
solving contexts do not exist in isolation.
The above points can be summarised as an independent learner who can think critically and analytically,
while also being able to work effectively with members of a team and identify and solve problems through
effective decision making. This is also the outcome of what educational research terms the "reformed"
approach rather than the "traditional" approach many educators are more accustomed to. Traditional
practices have their role and cannot be totally abandoned in favour of only reform practices. However,
in order to produce more independent and mathematical thinkers, the reform ideology needs to be more
embraced by educators within their instructional behaviour. Here is a table that can guide you to identify
your dominant instructional practice and try to assist you in adjusting it (if necessary) to be more balanced
and in line with the reform approach being suggested by the NCS.
Traditional Versus Reform Practices
Values
Traditional - values content, correctness of learn-
ers' responses and mathematical validity of meth-
ods. Reform - values finding patterns, making
connections, communicating mathematically and
problem-solving.
Teaching Methods
Traditional - expository, transmission, lots of
drill and practice, step by step mastery of algo-
rithms. Reform - hands-on guided discovery meth-
ods, exploration, modelling. High level reasoning
processes are central.
Grouping Learners
Traditional - dominantly same grouping ap-
proaches. Reform - dominantly mixed grouping
and abilities.
Table 1.2
The subject of mathematics, by the nature of the discipline, provides ample opportunities to meet the
reformed objectives. In doing so, the definition of mathematics needs to be understood and embraced by
educators involved in the teaching and the learning of the subject. In research it has been well documented
that, as educators, our conceptions of what mathematics is, has an influence on our approach to the teaching
and learning of the subject.
Three possible views of mathematics can be presented. The instrumentalist view of mathematics assumes
the stance that mathematics is an accumulation of facts, rules and skills that need to be used as a means
to an end, without there necessarily being any relation between these components. The Platonist view
of mathematics sees the subject as a static but unified body of certain knowledge, in which mathematics
is discovered rather than created. The problem solving view of mathematics is a dynamic, continually
expanding and evolving field of human creation and invention that is in itself a cultural product. Thus
mathematics is viewed as a process of enquiry, not a finished product. The results remain constantly open to
revision. It is suggested that a hierarchical order exists within these three views, placing the instrumentalist
view at the lowest level and the problem solving view at the highest.
1.1.3 According to the NCS:
Mathematics is the study of quantity, structure, space and change. Mathematicians seek out patterns,
formulate new conjectures, and establish axiomatic systems by rigorous deduction from appropriately chosen
axioms and definitions. Mathematics is a distinctly human activity practised by all cultures, for thousands
of years. Mathematical problem solving enables us to understand the world (physical, social and economic)
around us, and, most of all, to teach us to think creatively.
This corresponds well to the problem solving view of mathematics and may challenge some of our instru-
mentalist or Platonistic views of mathematics as a static body of knowledge of accumulated facts, rules and
skills to be learnt and applied. The NCS is trying to discourage such an approach and encourage mathemat-
ics educators to dynamically and creatively involve their learners as mathematicians engaged in a process of
study, understanding, reasoning, problem solving and communicating mathematically.
Below is a check list that can guide you in actively designing your lessons in an attempt to embrace the
definition of mathematics from the NCS and move towards a problem solving conception of the subject.
Adopting such an approach to the teaching and learning of mathematics will in turn contribute to the
intended curriculum being properly implemented and attained through the quality of learners coming out of
the education system.
CHAPTER 1. OVERVIEW
Practice
Example
Learners engage in solving contextual problems re-
lated to their lives that require them to interpret
a problem and then find a suitable mathematical
solution.
Learners are asked to work out which bus service
is the cheapest given the fares they charge and the
distance they want to travel.
Learners engage in solving problems of a purely
mathematical nature, which require higher order
thinking and application of knowledge (non-routine
problems).
Learners are required to draw a graph; they have
not yet been given a specific technique on how to
draw (for example a parabola), but have learnt to
use the table method to draw straight-line graphs.
Learners are given opportunities to negotiate mean-
ing.
Learners discuss their understanding of concepts
and strategies for solving problems with each other
and the educator.
Learners are shown and required to represent situa-
tions in various but equivalent ways (mathematical
modelling) .
Learners represent data using a graph, a table and
a formula to represent the same data.
Learners individually do mathematical investiga-
tions in class, guided by the educator where nec-
essary.
Each learner is given a paper containing the math-
ematical problem (for instance to find the number
of prime numbers less than 50) that needs to be in-
vestigated and the solution needs to be written up.
Learners work independently.
Learners work together as a group/team to investi-
gate or solve a mathematical problem.
A group is given the task of working together to
solve a problem that requires them investigating
patterns and working through data to make con-
jectures and find a formula for the pattern.
Learners do drill and practice exercises to consoli-
date the learning of concepts and to master various
skills.
Completing an exercise requiring routine proce-
dures.
Learners are given opportunities to see the inter-
relatedness of the mathematics and to see how the
different outcomes are related and connected.
While learners work through geometry problems,
they are encouraged to make use of algebra.
Learners are required to pose problems for their ed-
ucator and peer learners.
Learners are asked to make up an algebraic word
problem (for which they also know the solution) for
the person sitting next to them to solve.
Table 1.3
1.1.4 Outcomes
Summary of topics, outcomes and their relevance:
1. Functions -
- linear, quadratic, exponential, rational
continued on next page
Outcome
Relevance
10.1.1
Relationships between variables
in terms of graphical, verbal and
symbolic representations of func-
tions (tables, graphs, words and
formulae) .
Functions form a core part of
learners' mathematical
understanding and reasoning
processes in algebra. This is also
an excellent opportunity for
contextual mathematical
modelling questions.
10.1.2
Generating graphs and generalis-
ing effects of parameters of ver-
tical shifts and stretches and re-
flections about the x-axis.
10.1.3
Problem solving and graph work
involving prescribed functions.
Table 1.4
2. Number Patterns,
Sequences and Series
Outcome
Relevance
10.2.1
Number patterns with constant
difference.
Much of mathematics revolves
around the identification of pat-
terns.
Table 1.5
3. Finance, Growth and Decay
Outcome
Relevance
10.3.1
Use simple and compound
growth formulae.
The mathematics of finance is
very relevant to daily and
long-term financial decisions
learners will need to make in
terms of investing, taking loans,
10.3.2
Implications of fluctuating ex-
change rates.
Table 1.6
saving and understanding
exchange rates and their
influence more globally.
4. Algebra
Outc
Relevance
10.4.1
Identifying and converting forms
of rational numbers. Working
with simple surds that are not
rational.
Algebra provides the basis for
mathematics learners to move
from numerical calculations to
generalising operations,
simplifying expressions, solving
equatKSHlftaHffiflih^ffiBg and
inequalities in solving contextual
problems.
CHAPTER 1. OVERVIEW
10.4.2
Working with laws of inte-
gral exponents. Establish between
which two integers a simple surd
lies. Appropriately rounding off
real numbers.
10.4.3
Manipulating and simplifying al-
gebraic expressions (including
multiplication and factorisation).
10.4.4
Solving linear, quadratic and ex-
ponential equations. Solving lin-
ear inequalities in one and two
variables algebraically and graph-
ically.
Table 1.7
5. Differential Calculus
Outcome
Relevance
10.5.1
Investigate average rate of change
The central aspect of rate of
between two independent values
change to differential calculus is a
of a function.
basis to further understanding of
limits, gradients and calculations
and formulae necessary for work
in engineering fields, e.g. design-
ing roads, bridges etc.
Table 1.8
6. Probability
Outcome
Relevance
10.6.1
Compare relative frequency and
This topic is helpful in developing
theoretical probability.Use Venn
good logical reasoning in learn-
diagrams to solve probability
ers and for educating them in
problems. Mutually exclusive and
terms of real-life issues such as
complementary events. Identity
gambling and the possible pitfalls
for any two events A and B.
thereof.
Table 1.9
7. Euclidean Geometry and Measurement
Outcome
Relevance
10.7.1
Investigate, form and try to prove
conjectures about properties of
special triangles, quadrilaterals
and other polygons. Disprove
false conjectures using counter-
examples. Investigate alternative
definitions of various polygons.
The thinking processes and
mathematical skills of proving
conjectures and identifying false
conjectures is more the relevance
here than the actual content
studied. The surface area and
volume content studied in
real-life contexts of designing
kitchens, tiling and painting
rooms, designing packages, etc.
is relevant to the current and
10.7.2
Solve problems involving surface
area and volumes of solids and
combinations thereof.
Table 1.10
future lives of learners.
8. Trigonometry
Outcome
Relevance
10.8.1
Definitions of trig func-
tions. Derive values for special
angles. Take note of names for
reciprocal functions.
Trigonometry has several uses
within society, including within
navigation, music, geographical
locations and building design
and construction.
10.8.2
Solve problems in 2 dimensions.
10.8.3
Extend definition of basic trig
functions to all four quadrants
and know graphs of these func-
tions.
10.8.4
Investigate and know the effects
of a and q on the graphs of basic
trig functions.
Table 1.11
9. Analytical Geometry
Outcome
Relevance
10.9.1
Represent geometric figures on a
This section provides a further
Cartesian coordinate system. For
application point for learners' al-
any two points, derive and apply
gebraic and trigonometric inter-
formula for calculating distance,
action with the Cartesian plane.
gradient of line segment and co-
Artists and design and layout in-
ordinates of mid-point.
dustries often draw on the con-
tent and thought processes of this
mathematical topic.
Table 1.12
CHAPTER 1. OVERVIEW
10. Statistics
Outcome
Relevance
10.10.1
Collect, organise and interpret
univariate numerical data to de-
termine mean, median, mode,
percentiles, quartiles, deciles, in-
terquartile and semi-interquartile
range.
Citizens are daily confronted
with interpreting data presented
from the media. Often this data
may be biased or misrepresented
within a certain context. In any
type of research, data collection
and handling is a core feature.
This topic also educates learners
to become more socially and
10.10.2
Identify possible sources of bias
and errors in measurements.
Table 1.13
politically educated with regards
to the media.
Mathematics educators also need to ensure that the following important specific aims and general prin-
ciples are applied in mathematics activities across all grades:
• Calculators should only be used to perform standard numerical computations and verify calculations
done by hand.
• Real-life problems should be incorporated into all sections to keep mathematical modelling as an
important focal point of the curriculum.
• Investigations give learners the opportunity to develop their ability to be more methodical, to generalise
and to make and justify and/or prove conjectures.
• Appropriate approximation and rounding skills should be taught and continuously included and en-
couraged in activities.
• The history of mathematics should be incorporated into projects and tasks where possible, to illustrate
the human aspect and developing nature of mathematics.
• Contextual problems should include issues relating to health, social, economic, cultural, scientific,
political and environmental issues where possible.
• Conceptual understanding of when and why should also feature in problem types.
• Mixed ability teaching requires educators to challenge able learners and provide remedial support where
necessary.
• Misconceptions exposed by assessment need to be dealt with and rectified by questions designed by
educators.
• Problem solving and cognitive development should be central to all mathematics teaching and learning
so that learners can apply the knowledge effectively.
Chapter 2
Assessment and Assessment Support 1
2.1 Assessment
"Educator assessment is part of everyday teaching and learning in the classroom. Educators discuss with
learners, guide their work, ask and answer questions, observe, help, encourage and challenge. In addition,
they mark and review written and other kinds of work. Through these activities they are continually finding
out about their learners' capabilities and achievements. This knowledge then informs plans for future work.
It is this continuous process that makes up educator assessment. It should not be seen as a separate activity
necessarily requiring the use of extra tasks or tests."
As the quote above suggests, assessment should be incorporated as part of the classroom practice, rather
than as a separate activity. Research during the past ten years indicates that learners get a sense of what
they do and do not know, what they might do about this and how they feel about it, from frequent and regu-
lar classroom assessment and educator feedback. The educator's perceptions of and approach to assessment
(both formal and informal assessment) can have an influence on the classroom culture that is created with
regard to the learners' expectations of and performance in assessment tasks. Literature on classroom assess-
ment distinguishes between two different purposes of assessment; assessment of learning and assessment for
learning.
Assessment of learning tends to be a more formal assessment and assesses how much learners have learnt
or understood at a particular point in the annual teaching plan. The NCS provides comprehensive guidelines
on the types of and amount of formal assessment that needs to take place within the teaching year to make up
the school-based assessment mark. The school-based assessment mark contributes 25% of the final percentage
of a learner's promotion mark, while the end-of-year examination constitutes the other 75% of the annual
promotion mark. Learners are expected to have 7 formal assessment tasks for their school-based assessment
mark. The number of tasks and their weighting in the Grade 10 Mathematics curriculum is summarised
below:
Tasks
Weight (%)
Term 1
School-Based
Test
Project/Investigation
10
20
Assessment
continued on next page
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10
CHAPTER 2. ASSESSMENT AND ASSESSMENT SUPPORT
Term 2
Assignment /TestExamin£,tikfti
30
Term 3
TestTest
10
10
Term 4
Test
10
School-Based Assessment Mark
100
School-Based Assessment Mark (as a % of Promo-
tion Mark)
25
End-of-Year Examination
75
Promotion Mark
100
Table 2.1
The following provides a brief explanation of each of the assessment tasks included in the assessment
programme above.
2.1.1 Tests
All mathematics educators are familiar with this form of formal assessment. Tests include a variety of
items/questions covering the topics that have been taught prior to the test. The new NCS also stipulates
that mathematics tests should include questions that cover the following four types of cognitive levels in the
stipulated weightings:
Cognitive Levels
Description
Weighting (%)
Knowledge
Estimation and appropriate
rounding of numbers. Proofs of
prescribed theorems. Derivation
of formulae. Straight re-
call. Identification and direct
use of formula on informa-
tion sheet (no changing of the
subject). Use of mathemati-
cal facts. Appropriate use of
mathematical vocabulary.
20
continued on next page
11
Routine Procedures
Perform well known proce-
dures. Simple applications and
calculations. Derivation from
given information. Identification
and use (including changing
the subject) of correct for-
mula. Questions generally similar
to those done in class.
45
Complex Procedures
Problems involve complex cal-
culations and/or higher reason-
ing. There is often not an obvious
route to the solution. Problems
need not be based on real world
context. Could involve making
significant connections between
different representations. Require
conceptual understanding.
25
Problem Solving
Unseen, non-routine problems
(which are not necessarily dif-
ficult). Higher order understand-
ing and processes are often in-
volved. Might require the ability
to break the problem down into
its constituent parts.
10
Table 2.2
The breakdown of the tests over the four terms is summarised from the NCS assessment programme as
follows:
Term 1: One test of at least 50 marks, and one hour or two/three tests of at least 40 minutes each.
Term 2: Either one test (of at least 50 marks) or an assignment.
Term 3: Two tests, each of at least 50 marks and one hour.
Term 4: One test of at least 50 marks .
2.1.2 Projects / Investigations
Investigations and projects consist of open-ended questions that initiate and expand thought processes.
Acquiring and developing problem-solving skills are an essential part of doing investigations and projects.
These tasks provide learners with the opportunity to investigate, gather information, tabulate results, make
conjectures and justify or prove these conjectures. Examples of investigations and projects and possible
marking rubrics are provided in the next section on assessment support. The NCS assessment programme
indicates that only one project or investigation (of at least 50 marks) should be included per year. Although
the project/investigation is scheduled in the assessment programme for the first term, it could also be done
in the second term.
2.1.3 Assignments
The NCS includes the following tasks as good examples of assignments:
• Open book test
• Translation task
• Error spotting and correction
12
CHAPTER 2. ASSESSMENT AND ASSESSMENT SUPPORT
• Shorter investigation
• Journal entry
• Mind-map (also known as a metacog)
• Olympiad (first round)
• Mathematics tutorial on an entire topic
• Mathematics tutorial on more complex/problem solving questions
The NCS assessment programme requires one assignment in term 2 (of at least 50 marks) which could also
be a combination of some of the suggested examples above. More information on these suggested examples
of assignments and possible rubrics are provided in the following section on assessment support.
2.1.4 Examinations
Educators are also all familiar with this summative form of assessment that is usually completed twice a
year: mid-year examinations and end-of-year examinations. These are similar to the tests but cover a wider
range of topics completed prior to each examination. The NCS stipulates that each examination should
also cover the four cognitive levels according to their recommended weightings as summarised in the section
above on tests. The following table summarises the requirements and information from the NCS for the two
examinations.
Examination
Marks
Breakdown
Content and Mark
Distribution
Mid- Year Exam
100 50 + 50
One paper: 2 hours
or
Two papers: each of 1
hour
Topics completed
End-of-Year Exam
100 +
Paper 1: 2 hours
Number patterns (±10)
Algebraic expressions,
equations and inequali-
ties (±25)
Functions (±35)
Exponents (±10)
Finance (±10)
Probability (±10)
continued on next page
13
100
Paper 2: 2 hours
Trigonometry (±45)
Analytical geometry
(±15)
Euclidean geometry and
measurement (±25)
Statistics (±15)
Table 2.3
In the annual teaching plan summary of the NCS in Mathematics for Grade 10, the pace setter section
provides a detailed model of the suggested topics to be covered each week of each term and the accompanying
formal assessment.
Assessment for learning tends to be more informal and focuses on using assessment in and of daily
classroom activities that can include:
• Marking homework
• Baseline assessments
• Diagnostic assessments
• Group work
• Class discussions
• Oral presentations
• Self-assessment
• Peer-assessment
These activities are expanded on in the next section on assessment support and suggested marking rubrics
are provided. Where formal assessment tends to restrict the learner to written assessment tasks, the informal
assessment is necessary to evaluate and encourage the progress of the learners in their verbal mathematical
reasoning and communication skills. It also provides a less formal assessment environment that allows learners
to openly and honestly assess themselves and each other, taking responsibility for their own learning, without
the heavy weighting of the performance (or mark) component. The assessment for learning tasks should be
included in the classroom activities at least once a week (as part of a lesson) to ensure that the educator is
able to continuously evaluate the learners' understanding of the topics covered as well as the effectiveness,
and identify any possible deficiencies in his or her own teaching of the topics.
2,1,5 Assessment Support
A selection of explanations, examples and suggested marking rubrics for the assessment of learning (formal)
and the assessment for learning (informal) forms of assessment discussed in the preceding section are provided
in this section.
2.1.5.1 Baseline Assessment
Baseline assessment is a means of establishing:
• What prior knowledge a learner possesses
• What the extent of knowledge is that they have regarding a specific learning area
• The level they demonstrate regarding various skills and applications
• The learner's level of understanding of various learning areas
It is helpful to educators in order to assist them in taking learners from their individual point of departure
to a more advanced level and to thus make progress. This also helps avoid large "gaps" developing in the
learners' knowledge as the learner moves through the education system. Outcomes-based education is a more
learner-centered approach than we are used to in South Africa, and therefore the emphasis should now be
on the level of each individual learner rather than that of the whole class.
14 CHAPTER 2. ASSESSMENT AND ASSESSMENT SUPPORT
The baseline assessments also act as a gauge to enable learners to take more responsibility for their own
learning and to view their own progress. In the traditional assessment system, the weaker learners often drop
from a 40% average in the first term to a 30% average in the fourth term due to an increase in workload,
thus demonstrating no obvious progress. Baseline assessment, however, allows for an initial assigning of
levels which can be improved upon as the learner progresses through a section of work and shows greater
knowledge, understanding and skill in that area.
2.1.5.2 Diagnostic Assessments
These are used to specifically find out if any learning difficulties or problems exist within a section of work
in order to provide the learner with appropriate additional help and guidance. The assessment helps the
educator and the learner identify problem areas, misunderstandings, misconceptions and incorrect use and
interpretation of notation.
Some points to keep in mind:
• Try not to test too many concepts within one diagnostic assessment.
• Be selective in the type of questions you choose.
• Diagnostic assessments need to be designed with a certain structure in mind. As an educator, you
should decide exactly what outcomes you will be assessing and structure the content of the assessment
accordingly.
• The assessment is marked differently to other tests in that the mark is not the focus but rather the
type of mistakes the learner has made.
An example of an understanding rubric for educators to record results is provided below:
- indicates that the learner has not grasped the concept at all and that there appears to be a fundamental
mathematical problem.
1 - indicates that the learner has gained some idea of the content, but is not demonstrating an under-
standing of the notation and concept.
2 - indicates evidence of some understanding by the learner but further consolidation is still required.
3 - indicates clear evidence that the learner has understood the concept and is using the notation correctly.
An example of a diagnostic assessment to evaluate learners' proficiency in calculator skills is provided
below. There is a component of self-assessment as well as a component on educator assessment and how to
group the various questions to diagnose any gaps or problems with learners' calculator skills.
Calculator worksheet - diagnostic skills assessment
Question 1
Calculate:
a) 242 + 63
b) 2 - 36 x (114 + 25)
c) V144 + 25
d) ^729
e) -312 + 6 + 879 - 321 + 18 901
Question 2
Calculate:
a) ! + ! = ___
b , )2 Ms = -
c - 2 §3+t = -
d) 4 - § x § = .
e MlO 9/ • 5
f)2x(|) 2 -(i) =
1 9 4_
4 16 —
Self- Assessment Rubric: Name:
15
Question
Answer
V
X
If X, write down sequence of keys pressed
la)
lb)
1c)
Id)
le)
Subtotal
2a)
2b)
2c)
2d)
2e)
2f)
2g)
Subtotal
Total
Table 2.4
Educator Assessment Rubric:
Type of Skill
Competent
Needs Practice
Problem
Raising to a Power
Finding a Root
Calculations with Fractions
Brackets and Order of Operations
Estimation and Mental Control
Table 2.5
Guidelines for Calculator Skills Assessment:
Type of Skill
Sub-Division
Questions
Raising to a Power
Squaring and cubing
Higher order powers
la, 2f
lb
continued on next page
16
CHAPTER 2. ASSESSMENT AND ASSESSMENT SUPPORT
Finding a Root
Square and cube roots
lc, 2g
Higher order roots
Id
Calculations with Fractions
Basic operations
2a, 2d
Mixed numbers
2b, 2c
Negative numbers
le, 2c
Squaring fractions
2f
Square rooting fractions
2g
Brackets and Order of Opera-
Correct use of brackets or order
lb, lc, 2e,
2f, 2g
tions
of operations
Estimation and Mental Control
Overall
All
Table 2.6
Suggested guideline to allocation of overall levels
Level 1
• Learner is able to do basic operations on calculator.
• Learner is able to do simple calculations involving fractions.
• Learner does not display sufficient mental estimation and control techniques.
Level 2
• Learner is able to do basic operations on calculator.
• Learner is able to square and cube whole numbers as well as find square and cube roots of numbers.
• Learner is able to do simple calculations involving fractions as well as correctly execute calculations
involving mixed numbers.
• Learner displays some degree of mental estimation awareness.
Level 3
Learner is able to do basic operations on calculator.
Learner is able to square and cube rational numbers as well as find square and cube roots of numbers.
Learner is also able to calculate higher order powers and roots.
Learner is able to do simple calculations involving fractions as well as correctly execute calculations
involving mixed numbers.
Learner works correctly with negative numbers.
Learner is able to use brackets in certain calculations but has still not fully understood the order of
operations that the calculator has been programmed to execute, hence the need for brackets.
Learner is able to identify possible errors and problems in their calculations but needs assistance solving
the problem.
Level 4
Learner is able to do basic operations on calculator.
Learner is able to square and cube rational numbers as well as find square and cube roots.
Learner is also able to calculate higher order powers and roots.
Learner is able to do simple calculations involving fractions as well as correctly execute calculations
involving mixed numbers.
Learner works correctly with negative numbers.
Learner is able to work with brackets correctly and understands the need and use of brackets and the
"= key" in certain calculations due to the nature of a scientific calculator.
Learner is able to identify possible errors and problems in their calculations and to find solutions to
these in order to arrive at a "more viable" answer.
17
2,1,6 Other Short Diagnostic Tests
These are short tests that assess small quantities of recall knowledge and application ability on a day-to-day
basis. Such tests could include questions on one or a combination of the following:
• Definitions
• Theorems
• Riders (geometry)
• Formulae
• Applications
• Combination questions
Geometry
l.Points A (-5 ; -3), B (-1 ; 2) and C (9 ; -6) are the vertices of AABC.
a) Calculate the gradients of AB and BC and hence show that angle ABC is equal to 90 . (5)
b) State the distance formula and use it to calculate the lengths of the sides AB, BC and AC of AABC.
(Leave your answers in surd form). (5)
Algebra
1. Write down the formal definition of an exponent as well as the exponent laws for integral exponents.
(6)
2. Simplify: *jg£ x ^ (4)
Trigonometry
1. A jet leaves an airport and travels 578 km in a direction of 50 E of N. The pilot then changes direction
and travels 321 km 10 °W of N.
a) How far away from the airport is the jet? (To the nearest kilometre) (5)
b) Determine the jet's bearing from the airport. (5)
2.1.6.1 Exercises
This entails any work from the textbook or other source that is given to the learner, by the educator, to
complete either in class or at home. Educators should encourage learners not to copy each other's work and
be vigilant when controlling this work. It is suggested that such work be marked/controlled by a check list
(below) to speed up the process for the educator.
The marks obtained by the learner for a specific piece of work need not be based on correct and/or
incorrect answers but preferably on the following:
• the effort of the learner to produce answers.
• the quality of the corrections of work that was previously incorrect.
• the ability of the learner to explain the content of some selected examples (whether in writing or orally) .
The following rubric can be used to assess exercises done in class or as homework:
Critt
Performance Indicators
continued on next page
IS
CHAPTER 2. ASSESSMENT AND ASSESSMENT SUPPORT
Work Done
2 All the work
1 Partially completed
No work done
Work Neatly Done
2 Work neatly done
1 Some work not neatly
done
Messy and muddled
Corrections Done
2 All corrections done
consistently
1 At least half of the cor-
rections done
No corrections done
Correct Mathematical
Method
2 Consistently
1 Sometimes
Never
Understanding of Math-
ematical Techniques
and Processes
2 Can explain concepts
and processes precisely
1 Explanations are am-
biguous or not focused
Explanations are con-
fusing or irrelevant
Table 2.7
2.1.6.2 Journal entries
A journal entry is an attempt by a learner to express in the written word what is happening in Mathematics.
It is important to be able to articulate a mathematical problem, and its solution in the written word.
This can be done in a number of different ways:
• Today in Maths we learnt
• Write a letter to a friend, who has been sick, explaining what was done in class today.
• Explain the thought process behind trying to solve a particular maths problem, e.g. sketch the graph
of y = x 2 2x 2 + land explain how to sketch such a graph.
• Give a solution to a problem, decide whether it is correct and if not, explain the possible difficulties
experienced by the person who wrote the incorrect solution.
A journal is an invaluable tool that enables the educator to identify any mathematical misconceptions of the
learners. The marking of this kind of exercise can be seen as subjective but a marking rubric can simplify
the task.
The following rubric can be used to mark journal entries. The learners must be given the marking rubric
before the task is done.
Task
Competent (2
Marks)
Still Developing (1
Mark)
Not Yet Developed
(0 Marks)
Completion in Time
Limit?
Correctness of the Ex-
planation?
Correct and Relevant
use of Mathematical
Language?
continued on next page
19
Is the Mathematics Cor-
rect?
Has the Concept Been
Interpreted Correctly?
Table 2.8
2.1.6.3 Translations
Translations assess the learner's ability to translate from words into mathematical notation or to give an
explanation of mathematical concepts in words. Often when learners can use mathematical language and
notation correctly, they demonstrate a greater understanding of the concepts.
For example:
Write the letter of the correct expression next to the matching number:
x increased by 10 a) xy
The product of x and y b) x 2
The sum of a certain number and c) x 2
double that number d) 29x
Half of a certain number multiplied by itself e) jx2
Two less than x f ) x + x + 2
A certain number multiplied by itself g) x 2
Two consecutive even numbers h) x 29
x + x + x + to 29 terms i) x + 2x
x.x.x.x.x to 29 factors j) x + 10
A certain number divided by 2
2.1.6.4 Group Work
One of the principles in the NCS is to produce learners who are able to work effectively within a group.
Learners generally find this difficult to do. Learners need to be encouraged to work within small groups.
Very often it is while learning under peer assistance that a better understanding of concepts and processes
is reached. Clever learners usually battle with this sort of task, and yet it is important that they learn how
to assist and communicate effectively with other learners.
2.1.6.5 Mind Maps or Metacogs
A metacog or "mind map" is a useful tool. It helps to associate ideas and make connections that would
otherwise be too unrelated to be linked. A metacog can be used at the beginning or end of a section of work
in order to give learners an overall perspective of the work covered, or as a way of recalling a section already
completed. It must be emphasised that it is not a summary. Whichever way you use it, it is a way in which
a learner is given the opportunity of doing research in a particular field and can show that he/she has an
understanding of the required section.
This is an open book form of assessment and learners may use any material they feel will assist them.
It is suggested that this activity be practised, using other topics, before a test metacog is submitted for
portfolio assessment purposes.
On completion of the metacog, learners must be able to answer insightful questions on the metacog. This
is what sets it apart from being just a summary of a section of work. Learners must refer to their metacog
when answering the questions, but may not refer to any reference material. Below are some guidelines to
give to learners to adhere to when constructing a metacog as well as two examples to help you get learners
started. A marking rubric is also provided. This should be made available to learners before they start
20
CHAPTER 2. ASSESSMENT AND ASSESSMENT SUPPORT
constructing their metacogs. On the next page is a model question for a metacog, accompanied by some
sample questions that can be asked within the context of doing a metacog about analytical geometry.
A basic metacog is drawn in the following way:
• Write the title/topic of the subject in the centre of the page and draw a circle around it.
• For the first main heading of the subject, draw a line out from the circle in any direction, and write
the heading above or below the line.
• For sub-headings of the main heading, draw lines out from the first line for each subheading and label
each one.
• For individual facts, draw lines out from the appropriate heading line.
Metacogs are one's own property. Once a person understands how to assemble the basic structure they can
develop their own coding and conventions to take things further, for example to show linkages between facts.
The following suggestions may assist educators and learners to enhance the effectiveness of their metacogs:
• Use single words or simple phrases for information. Excess words just clutter the metacog and take
extra time to write down.
• Print words - joined up or indistinct writing can be more difficult to read and less attractive to look
at.
• Use colour to separate different ideas - this will help your mind separate ideas where it is necessary,
and helps visualisation of the metacog for easy recall. Colour also helps to show organisation.
• Use symbols and images where applicable. If a symbol means something to you, and conveys more
information than words, use it. Pictures also help you to remember information.
• Use shapes, circles and boundaries to connect information - these are additional tools to help show
the grouping of information.
Use the concept of analytical geometry as your topic and construct a mind map (or metacog) containing all
the information (including terminology, definitions, formulae and examples) that you know about the topic
of analytical geometry.
Possible questions to ask the learner on completion of their metacog:
• Briefly explain to me what the mathematics topic of analytical geometry entails.
• Identify and explain the distance formula, the derivation and use thereof for me on your metacog.
• How does the calculation of gradient in analytical geometry differ (or not) from the approach used to
calculate gradient in working with functions?
A suggested simple rubric for marking a metacog:
Task
Competent (2
Marks)
Still Developing (1
Mark)
Not Yet Developed
(0 Marks)
Completion in Time
Limit
Main Headings
continued on next page
21
Correct Theory (Formu-
lae, Definitions, Termi-
nology etc.)
Explanation
"Readability"
Table 2.9
10 marks for the questions, which are marked using the following scale:
- no attempt or a totally incorrect attempt has been made
1 - a correct attempt was made, but the learner did not get the correct answer
2 - a correct attempt was made and the answer is correct
2.1.6.6 Investigations
Investigations consist of open-ended questions that initiate and expand thought processes. Acquiring and
developing problem-solving skills are an essential part of doing investigations.
It is suggested that 2-3 hours be allowed for this task. During the first 30 - 45 minutes learners could
be encouraged to talk about the problem, clarify points of confusion, and discuss initial conjectures with
others. The final written-up version should be done individually though and should be approximately four
pages.
Assessing investigations may include feedback/ presentations from groups or individuals on the results
keeping the following in mind:
• following of a logical sequence in solving the problems
• pre-knowledge required to solve the problem
• correct usage of mathematical language and notation
• purposefulness of solution
• quality of the written and oral presentation
Some examples of suggested marking rubrics are included on the next few pages, followed by a selection of
topics for possible investigations.
The following guidelines should be provided to learners before they begin an investigation:
General Instructions Provided to Learners
1. You may choose any one of the projects/investigations given (see model question on investigations)
2. You should follow the instructions that accompany each task as these describe the way in which the
final product must be presented.
3. You may discuss the problem in groups to clarify issues, but each individual must write-up their own
version.
4. Copying from fellow learners will cause the task to be disqualified.
5. Your educator is a resource to you, and though they will not provide you with answers / solutions,
they may be approached for hints.
The Presentation
The investigation is to be handed in on the due date, indicated to you by your educator. It should have
as a minimum:
• A description of the problem.
• A discussion of the way you set about dealing with the problem.
• A description of the final result with an appropriate justification of its validity.
• Some personal reflections that include mathematical or other lessons learnt, as well as the feelings
experienced whilst engaging in the problem.
22
CHAPTER 2. ASSESSMENT AND ASSESSMENT SUPPORT
• The written-up version should be attractively and neatly presented on about four A4 pages.
• Whilst the use of technology is encouraged in the presentation, the mathematical content and processes
must remain the major focus.
Below are some examples of possible rubrics to use when marking investigations:
Example 1:
Level of Performance
Criteria
4
• Contains a complete response.
• Clear, coherent, unambiguous and elegant ex-
planation.
• Includes clear and simple diagrams where ap-
propriate.
• Shows understanding of the question's math-
ematical ideas and processes.
• Identifies all the important elements of the
question.
• Includes examples and counter examples.
• Gives strong supporting arguments.
• Goes beyond the requirements of the problem.
3
• Contains a complete response.
• Explanation less elegant, less complete.
• Shows understanding of the question's math-
ematical ideas and processes.
• Identifies all the important elements of the
question.
• Does not go beyond the requirements of the
problem.
continued on next page
23
2
• Contains an incomplete response.
• Explanation is not logical and clear.
• Shows some understanding of the question's
mathematical ideas and processes.
• Identifies some of the important elements of
the question.
• Presents arguments, but incomplete.
• Includes diagrams, but inappropriate or un-
clear.
1
• Contains an incomplete response.
• Omits significant parts or all of the question
and response.
• Contains major errors.
• Uses inappropriate strategies.
• No visible response or attempt
Table 2.10
2.1.6.7 Orals
An oral assessment involves the learner explaining to the class as a whole, a group or the educator his or her
understanding of a concept, a problem or answering specific questions. The focus here is on the correct use
of mathematical language by the learner and the conciseness and logical progression of their explanation as
well as their communication skills.
Orals can be done in a number of ways:
• A learner explains the solution of a homework problem chosen by the educator.
• The educator asks the learner a specific question or set of questions to ascertain that the learner
understands, and assesses the learner on their explanation.
• The educator observes a group of learners interacting and assesses the learners on their contributions
and explanations within the group.
• A group is given a mark as a whole, according to the answer given to a question by any member of a
group.
An example of a marking rubric for an oral:
1 - the learner has understood the question and attempts to answer it
2 - the learner uses correct mathematical language
2 - the explanation of the learner follows a logical progression
2 - the learner's explanation is concise and accurate
2 - the learner shows an understanding of the concept being explained
1 - the learner demonstrates good communication skills
Maximum mark =10
An example of a peer-assessment rubric for an oral:
My name:
Name of person I am assessing:
24
CHAPTER 2. ASSESSMENT AND ASSESSMENT SUPPORT
Criteria
Mark awarded
Maximum Mark
Correct Answer
2
Clarity of Explanation
3
Correctness of Explanation
3
Evidence of Understanding
2
Total
10
Table 2.11
Chapter 3
Chapter Contexts 1
3.1 Chapter Contexts
3.1.1 Functions and Graphs
Functions form a core part of learners' mathematical understanding and reasoning processes in algebra. This
is also an excellent opportunity for contextual mathematical modelling questions.
3.1.2 Number Patterns
Much of mathematics revolves around the identification of patterns.
3.1.3 Finance, Growth and Decay
The mathematics of finance is very relevant to daily and long-term financial decisions learners will need to
take in terms of investing, taking loans, saving and understanding exchange rates and their influence more
globally.
3.1.4 Algebra
Algebra provides the basis for mathematics learners to move from numerical calculations to generalising
operations, simplifying expressions, solving equations and using graphs and inequalities in solving contextual
problems.
3.1.5 Products and Factors
Being able to multiply out and factorise are core skills in the process of simplifying expressions and solving
equations in mathematics.
3.1.6 Equations and Inequalities
If learners are to later work competently with functions and the graphing and interpretation thereof, their
understanding and skills in solving equations and inequalities will need to be developed.
lr This content is available online at <http://cnx.Org/content/m40352/l.l/>.
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26 CHAPTER 3. CHAPTER CONTEXTS
3.1.7 Estimating Surds
Estimation is an extremely important component within mathematics. It allows learners to work with
a calculator or present possible solutions while still being able to gauge how accurate and realistic their
answers may be. This is relevant for other subjects too. For example, a learner working in biology may need
to do a calculation to find the size of the average human kidney. An erroneous interpretation or calculation
may result in an answer of 900 m. Without estimation skills, the learner may not query the possibility of such
an answer and consider that it should rather be 9 cm. Estimating surds facilitates the further development
of this skill of estimation.
3.1.8 Exponentials
Exponential notation is a central part of mathematics in numerical calculations as well as algebraic reasoning
and simplification. It is also a necessary component for learners to understand and appreciate certain financial
concepts such as compound interest and growth and decay.
3.1.9 Irrational Numbers & Rounding Off
Identifying irrational numbers and knowing their estimated position on a number line or graph is an im-
portant part of any mathematical calculations and processes that move beyond the basic number system of
whole numbers and integers. Rounding off irrational numbers (such as the value of n) when needed allows
mathematics learners to work more efficiently with numbers that would otherwise be difficult to "pin down",
use and comprehend.
3.1.10 Rational Numbers
Once learners have grasped the basic number system of whole numbers and integers, it is vital that their
understanding of the numbers between integers is also expanded. This incorporates their dealing with frac-
tions, decimals and surds which form a central part of most mathematical calculations in real-life contextual
3.1.11 Differential Calculus: Average Gradient
The central aspect of rate of change to differential calculus is a basis to further understanding of limits,
gradients and calculations and formulae necessary for work in engineering fields, e.g. designing roads, bridges
etc.
3.1.12 Probability
This topic is helpful in developing good logical reasoning in learners and for educating them in terms of
real-life issues such as gambling and the possible pitfalls thereof.
3.1.13 Euclidean Geometry and Measurement
The thinking processes and mathematical skills of proving conjectures and identifying false conjectures is
more the relevance here than the actual content studied. The surface area and volume content studied in
real-life contexts of designing kitchens, tiling and painting rooms, designing packages, etc. is relevant to the
current and future lives of learners.
3.1.14 Trigonometry
Trigonometry has several uses within society, including within navigation, music, geographical locations and
building design and construction.
27
3.1.15 Analytical Geometry
This section provides a further application point for learners' algebraic and trigonometric interaction with the
Cartesian plane. Artists and design and layout industries often draw on the content and thought processes
of this mathematical topic.
3.1.16 Statistics
Citizens are daily confronted with interpreting data presented from the media. Often this data may be biased
or misrepresented within a certain context. In any type of research, data collection and handling is a core
feature. This topic also educates learners to become more socially and politically educated with regards to
the media.
28 CHAPTER 3. CHAPTER CONTEXTS
Chapter 4
On the Web, Everyone can be a Scientist 1
4.1 On the Web, Everyone can be a Scientist
Did you know that you can fold protein molecules, hunt for new planets around distant suns or simulate
how malaria spreads in Africa, all from an ordinary PC or laptop connected to the Internet? And you don't
need to be a certified scientist to do this. In fact some of the most talented contributors are teenagers. The
reason this is possible is that scientists are learning how to turn simple scientific tasks into competitive online
games.
This is the story of how a simple idea of sharing scientific challenges on the Web turned into a global
trend, called citizen cyberscience. And how you can be a scientist on the Web, too.
4.1.1 Looking for Little Green Men
A long time ago, in 1999, when the World Wide Web was barely ten years old and no one had heard of Google,
Facebook or Twitter, a researcher at the University of California at Berkeley, David Anderson, launched an
online project called SETI@home. SETI stands for Search for Extraterrestrial Intelligence. Looking for life
in outer space.
Although this sounds like science fiction, it is a real and quite reasonable scientific project. The idea is
simple enough. If there are aliens out there on other planets, and they are as smart or even smarter than
us, then they almost certainly have invented the radio already. So if we listen very carefully for radio signals
from outer space, we may pick up the faint signals of intelligent life.
Exactly what radio broadcasts aliens would produce is a matter of some debate. But the idea is that if
they do, it would sound quite different from the normal hiss of background radio noise produced by stars
and galaxies. So if you search long enough and hard enough, maybe you'll find a sign of life.
It was clear to David and his colleagues that the search was going to require a lot of computers. More
than scientists could afford. So he wrote a simple computer program which broke the problem down into
smaller parts, sending bits of radio data collected by a giant radio-telescope to volunteers around the world.
The volunteers agreed to download a programme onto their home computers that would sift through the bit
of data they received, looking for signals of life, and send back a short summary of the result to a central
server in California.
The biggest surprise of this project was not that they discovered a message from outer space. In fact,
after over a decade of searching, no sign of extraterrestrial life has been found, although there are still vast
regions of space that have not been looked at. The biggest surprise was the number of people willing to help
such an endeavour. Over a million people have downloaded the software, making the total computing power
of SETI@home rival that of even the biggest supercomputers in the world.
1 This content is available online at <http://cnx.Org/content/m40353/l.l/>.
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30 CHAPTER 4. ON THE WEB, EVERYONE CAN BE A SCIENTIST
David was deeply impressed by the enthusiasm of people to help this project. And he realized that
searching for aliens was probably not the only task that people would be willing to help with by using the
spare time on their computers. So he set about building a software platform that would allow many other
scientists to set up similar projects. You can read more about this platform, called BOINC, and the many
different kinds of volunteer computing projects it supports today, at http://boinc.berkeley.edu/ 2 .
There's something for everyone, from searching for new prime numbers (PrimeGrid) to simulating the
future of the Earth's climate (ClimatePrediction.net). One of the projects, MalariaControl.net, involved
researchers from the University of Cape Town as well as from universities in Mali and Senegal.
The other neat feature of BOINC is that it lets people who share a common interest in a scientific topic
share their passion, and learn from each other. BOINC even supports teams - groups of people who put their
computer power together, in a virtual way on the Web, to get a higher score than their rivals. So BOINC is
a bit like Facebook and World of Warcraft combined - part social network, part online multiplayer game.
Here's a thought: spend some time searching around BOINC for a project you'd like to participate
in, or tell your class about.
4.1,2 You are a Computer, too
Before computers were machines, they were people. Vast rooms full of hundreds of government employees
used to calculate the sort of mathematical tables that a laptop can produce nowadays in a fraction of a
second. They used to do those calculations laboriously, by hand. And because it was easy to make mistakes,
a lot of the effort was involved in double-checking the work done by others.
Well, that was a long time ago. Since electronic computers emerged over 50 years ago, there has been
no need to assemble large groups of humans to do boring, repetitive mathematical tasks. Silicon chips can
solve those problems today far faster and more accurately. But there are still some mathematical problems
where the human brain excels.
Volunteer computing is a good name for what BOINC does: it enables volunteers to contribute computing
power of their PCs and laptops. But in recent years, a new trend has emerged in citizen cyberscience that
is best described as volunteer thinking. Here the computers are replaced by brains, connected via the
Web through an interface called eyes. Because for some complex problems - especially those that involve
recognizing complex patterns or three-dimensional objects - the human brain is still a lot quicker and more
accurate than a computer.
Volunteer thinking projects come in many shapes and sizes. For example, you can help to classify
millions of images of distant galaxies (GalaxyZoo), or digitize hand- written information associated with
museum archive data of various plant species (Herbaria@home). This is laborious work, which if left to
experts would take years or decades to complete. But thanks to the Web, it's possible to distribute images
so that hundreds of thousands of people can contribute to the search.
Not only is there strength in numbers, there is accuracy, too. Because by using a technique called valida-
tion - which does the same sort of double-checking that used to be done by humans making mathematical
tables - it is possible to practically eliminate the effects of human error. This is true even though each
volunteer may make quite a few mistakes. So projects like Planet Hunters have already helped astronomers
pinpoint new planets circling distant stars. The game Foldlt invites people to compete in folding protein
molecules via a simple mouse-driven interface. By finding the most likely way a protein will fold, volunteers
can help understand illnesses like Alzheimer's disease, that depend on how proteins fold.
Volunteer thinking is exciting. But perhaps even more ambitious is the emerging idea of volunteer
sensing: using your laptop or even your mobile phone to collect data - sounds, images, text you type in -
from any point on the planet, helping scientists to create global networks of sensors that can pick up the
first signs of an outbreak of a new disease (EpiCollect), or the initial tremors associated with an earthquake
(QuakeCatcher.net), or the noise levels around a new airport (NoiseTube).
There are about a billion PCs and laptops on the planet, but already 5 billion mobile phones. The
rapid advance of computing technology, where the power of a ten-year old PC can easily be packed into a
2 http://boinc. berkeley.edu/
31
smart phone today, means that citizen cyberscience has a bright future in mobile phones. And this means
that more and more of the world's population can be part of citizen cyberscience projects. Today there are
probably a few million participants in a few hundred citizen cyberscience initiatives. But there will soon be
seven billion brains on the planet. That is a lot of potential citizen cyberscientists.
You can explore much more about citizen cyberscience on the Web. There's a great list of all sorts of
projects, with brief summaries of their objectives, at http://distributedcomputing.info/ 3 . BBC Radio 4
produced a short series on citizen science http://www.bbc.co.uk/radio4/science/citizenscience.shtml 4 and
you can subscribe to a newsletter about the latest trends in this field at http://scienceforcitizens.net/ 5 .
The Citizen Cyberscience Centre, www.citizencyberscience.net 6 which is sponsored by the South African
Shuttleworth Foundation, is promoting citizen cyberscience in Africa and other developing regions.
3 http://distributedcomputing.info/
4 http:// www.bbc.co.uk/radio4/science/citizenscience.shtml
5u
3 http://scienceforcitizens.net/
3 http://www.citizencyberscience.net/
32 CHAPTER 4. ON THE WEB, EVERYONE CAN BE A SCIENTIST
Chapter 5
FullMarks User Guide 1
5.1 FullMarks User Guide
FullMarks can be accessed at: http://www.fullmarks.org.za/ 2 .
Siyavula offers an open online assessment bank called FullMarks, for the sharing and accessing of
curriculum-aligned test and exam questions with answers. This site enables educators to quickly set tests and
exam papers, by selecting items from the library and adding them to their test. Educators can then download
their separate test and memo which is ready for printing. FullMarks further offers educators the option of
capturing their learners' marks in order to view a selection of diagnostic reports on their performance.
To begin, you need to a create a free account by clicking on "sign up now" on the landing page. There is
one piece of administration you need to do to get started properly: when you log in for the first time, click
on your name on the top right. It will take you to your personal settings. You need to select Shuttleworth
Foundation as your metadata organisation to see the curriculum topics.
1 This content is available online at <http://cnx.Org/content/m40356/l.l/>.
2 http:// www.fullmarks.org.za/
33
34
CHAPTER 5. FULLMARKS USER GUIDE
5.1.1 What Can I do in FullMarks?
Access and
share questions
I
Create tests \ /
from questions/ *y
Create
class lists
Create \
scoresheets )
/Analyse learners'^
V performance J
Figure 5.1
5.1.2 How do I do Each of These?
5.1.2.1 Access and Share Questions
Sharing questions: use either the online editor or OpenOffice template which can be downloaded from the
website (Browse questions — > Contribute questions — » Import questions). Take your test/worksheet/other
question source. Break it up into the smallest sized individual questions that make sense, and use the
template style guide to style your page according to question/answer. Upload these questions or type them
up in the online editor (Browse questions — > Contribute questions — > Add questions). Do not include overall
question numbering but do include sub numbering if needed (e.g. la, 2c, etc.). Insert the mark and time
allocation, tag questions according to grade, subject i.e. a description of the question, and then select the
topics from the topic tree . Finalise questions so that they can be used in tests and accessed by other
FullMarks users.
Accessing questions: there are three ways to access questions in the database. Click on "Browse questions"
— > click on the arrow to the left of the grade, which opens out the subjects — * keep clicking on the arrows
to open the learning outcomes or, following the same process, instead of clicking on the arrow, click on the
grade — > now you can browse the full database of questions for all the subjects in that grade or, from the
35
landing page, click on "Browse questions" — > below the banner image click on "Find Questions" — > search by
topics, author (if you know a contributor), text or keywords e.g. GrlO mathematics functions and graphs.
5.1.2.2 Create Tests from Questions
So, you have all these bits of tests (i.e. many questions!), but what you really want is the actual test. How
do you do this? Well, you can simply click "add to test" on any question and then click on the "Tests" tab at
the top right of the page, and follow the simple instructions. Alternatively you can create a test by starting
with clicking on that same "Tests" tab, and add questions to your test that way. Once done, simply print
off the PDF file of the questions and the file for the memo. Issue your test, collect them once complete, and
mark them.
5.1.2.3 Create Class Lists
But now you are asking, how can I keep track of my classes? Is Johnny Brown in class A or B? Well, you
can make a class list by clicking on the "Class lists" tab at the top right of the page, and either import a
CSV file, or manually enter the relevant information for each class. Now you can issue tests to your classes,
and have a class list for each class. And what about capturing their marks?
5.1.2.4 Create Scoresheets
For each test you can create a scoresheet. Select the "Tests" tab at the top right, click on "Marks" below
the banner image, and select the test and follow the instructions to input their marks. You can then export
these as a CSV file for use in spreadsheets.
5.1.2.5 Analyse Learners' Performance
And finally, you can print out reports of class performance. Click on "Reports" at the top right of the page,
which opens various reports you can view. There are reports to see class performance, learner performance,
class performance per topic, class performance per question, learner strengths and weaknesses, and learner
progress.
So now you know how FullMarks works, we encourage you to make use of its simple functionality, and
let it help you save time setting tests and analysing learner marks!
36 CHAPTER 5. FULLMARKS USER GUIDE
Chapter 6
Rich Media
6.1 Rich Media
6.1.1 General Resources
Science education is about more than physics, chemistry and mathematics... It's about learning to think and
to solve problems, which are valuable skills that can be applied through all spheres of life. Teaching these
skills to our next generation is crucial in the current global environment where methodologies, technology
and tools are rapidly evolving. Education should benefit from these fast moving developments. In our
simplified model there are three layers to how technology can significantly influence your teaching and
teaching environment.
6.1.1.1 First Layer: Educator Collaboration
There are many tools that help educators collaborate more effectively. We know that communities of practice
are powerful tools for the refinement of methodology, content and knowledge as well as superb for providing
support to educators. One of the challenges facing community formation is the time and space to have
sufficient meetings to build real communities and exchanging practices, content and learnings effectively.
Technology allows us to streamline this very effectively by transcending space and time. It is now possible
to collaborate over large distances (transcending space) and when it is most appropriate for each individual
(transcending time) by working virtually (email, mobile, online etc.).
Our textbooks have been uploaded in their entirety to the Connexions website
(http://cnx.org/lenses/fhsst 2 ), making them easily accessible and adaptable, as they are under an
open licence, stored in an open format, based on an open standard, on an open-source platform, for free,
where everyone can produce their own books. Our textbooks are released under an open copyright license -
CC-BY. This Creative Commons By Attribution Licence allows others to legally distribute, remix, tweak,
and build upon our work, even commercially, as long as they credit us for the original creation. With them
being available on the Connexions website and due to the open copyright licence, learners and educators
are able to download, copy, share and distribute our books legally at no cost. It also gives educators the
freedom to edit, adapt, translate and contextualise them, to better suit their teaching needs.
Connexions is a tool where individuals can share, but more importantly communities can form around
the collaborative, online development of resources. Your community of educators can therefore:
• form an online workgroup around the textbook;
• make your own copy of the textbook;
• edit sections of your own copy;
1 This content is available online at <http://cnx.Org/content/m40357/l.l/>.
2 http://cnx.org/lenses/fhsst
37
38 CHAPTER 6. RICH MEDIA
• add your own content into the book or replace existing content with your content;
• use other content that has been shared on the platform in your own book;
• create your own notes / textbook / course material as a community.
Educators often want to share assessment items as this helps reduce workload, increase variety and im-
prove quality. Currently all the solutions to the exercises contained in the textbooks have been uploaded
onto our free and open online assessment bank called FullMarks (http://www.fullmarks.org.za/ 3 ), with
each exercise having a shortcode link to its solution on FullMarks. To access the solution simply go to
http://www.siyavula.com 4 , enter the shortcode, and you will be redirected to the solution on FullMarks.
FullMarks is similar to Connexions but is focused on the sharing of assessment items. FullMarks contains a
selection of test and exam questions with solutions, openly shared by educators. Educators can further search
and browse the database by subject and grade and add relevant items to a test. The website automatically
generates a test or exam paper with the corresponding memorandum for download.
By uploading all the end-of-chapter exercises and solutions to the open assessment bank, the larger
community of educators in South Africa are provided with a wide selection of items to use in setting their
tests and exams. More details about the use of FullMarks as a collaboration tool are included in the FullMarks
section.
6.1.1.2 Second Layer: Classroom Engagement
In spite of the impressive array of rich media open educational resources available freely online, such as
videos, simulations, exercises and presentations, only a small number of educators actively make use of
them. Our investigations revealed that the overwhelming quantity, the predominant international context,
and difficulty in correctly aligning them with the local curriculum level acts as deterrents. The opportunity
here is that, if used correctly, they can make the classroom environment more engaging.
Presentations can be a first step to bringing material to life in ways that are more compelling than are
possible with just a blackboard and chalk. There are opportunities to:
• create more graphical representations of the content;
• control timing of presented content more effectively;
• allow learners to relive the lesson later if constructed well;
• supplement the slides with notes for later use;
• embed key assessment items in advance to promote discussion; and
• embed other rich media like videos.
Videos have been shown to be potentially both engaging and effective. They provide opportunities to:
• present an alternative explanation;
• challenge misconceptions without challenging an individual in the class; and
• show an environment or experiment that cannot be replicated in the class which could be far away,
too expensive or too dangerous.
Simulations are also very useful and can allow learners to:
• have increased freedom to explore, rather than reproducing a fixed experiment or process;
• explore expensive or dangerous environments more effectively; and
• overcome implicit misconceptions.
We realised the opportunity for embedding a selection of rich media resources such as presentations, simu-
lations, videos and links into the online version of the FHSST books at the relevant sections. This will not
only present them with a selection of locally relevant and curriculum aligned resources, but also position
these resources within the appropriate grade and section. Links to these online resources are recorded in the
print or PDF versions of the books, making them a tour-guide or credible pointer to the world of online rich
media available.
3 http:// www.fullmarks.org.za/
4 http:// www.siyavula.com/
39
6.1.1.3 Third Layer: Beyond the Classroom
The internet has provided many opportunities for self-learning and participation which were never before
possible. There are huge stand-alone archives of videos like the Khan Academy which cover most Mathematics
for Grades 1-12 and Science topics required in FET. These videos, if not used in class, provide opportunities
for the learners to:
• look up content themselves;
• get ahead of class;
• independently revise and consolidate their foundation; and
• explore a subject to see if they find it interesting.
There are also many opportunities for learners to participate in science projects online as real participants.
Not just simulations or tutorials but real science so that:
• learners gain an appreciation of how science is changing;
• safely and easily explore subjects that they would never have encountered before university;
• contribute to real science (real international cutting edge science programmes);
• have the possibility of making real discoveries even from their school computer laboratory; and
• find active role models in the world of science.
In our book we've embedded opportunities to help educators and learners take advantage of all these re-
sources, without becoming overwhelmed at all the content that is available online.
6.1.2 Embedded Content
Throughout the books you will see the following icons:
Icon
Description
^
Aside: Provides additional information about con-
tent covered in the chapters, as well as for exten-
sions
Infc^^Hing
An interesting fact: These highlight interesting
information relevant to a particular section of the
chapter.
continued on next page
40
CHAPTER 6. RICH MEDIA
Definition: This icon indicates a definition.
Exercise: This indicates worked examples through-
out the book.
t
Tip: Helpful hints and tips appear throughout the
book, highlighting important information, things to
take note of, and areas where learners must exercise
caution.
FullMarks: This icon indicates that shortcodes
for FullMarks are present. Enter the shortcode into
http://www.siyavula.com 23 , and you will be redi-
rected to the solution on FullMarks, our free and
open online assessment bank. FullMarks can be ac-
cessed at: http://www.fullmarks.org.za/ 24
Presentation: This icon indicates that presen-
tations are in the chapter. Enter the shortcode
into http://www.siyavula.com 25 , and you will be
redirected to the presentation shared on SlideShare
by educators. SlideShare can be accessed at:
http://www.slideshare.net/ 26
continued on next page
41
Simulation: This icon indicates that simula-
tions are present. Enter the shortcode into
http://www.siyavula.com 27 , and you will be redi-
rected to the simulation online. An example is
Phet Simulations. The website can be accessed at:
http://phet.colorado.edu/ 28
Video: This icon indicates that videos
are present. Enter the shortcode into
http://www.siyavula.com 29 , and you will be
redirected to the video online. An example is
the Khan Academy videos. The website can be
accessed at: http://www.khanacademy.org/ 30
URL: This icon indicates that shortcodes are
present in the chapter and can be entered into
http://www.siyavula.com 31 , where you will be redi-
rected to the relevant website.
WWW
Table 6.1
23 http:
24 http:
25 http;
26 http;
27 http:
28 http:
29 http:
30 http:
31 http:
// www.siyavula.com/
// www.fullmarks.org.za/
// www.siyavula.com/
// www.slideshare.net/
// www.siyavula.com/
//phet. colorado.edu/
// www.siyavula.com/
// www.khanacademy.org/
// www.siyavula.com/
42 CHAPTER 6. RICH MEDIA
Chapter 7
Maths Blog Posts 1
7.1 Blog Posts
7.1.1 General Blogs
Teachers Monthly - Education News and Resources
• "We eat, breathe and live education! "
• "Perhaps the most remarkable yet overlooked aspect of the South African teaching community is its
enthusiastic, passionate spirit. Every day, thousands of talented, hard-working teachers gain new
insight from their work and come up with brilliant, inventive and exciting ideas. Teacher's Monthly
aims to bring teachers closer and help them share knowledge and resources.
• Our aim is twofold . . .
• To keep South African teachers updated and informed.
• To give teachers the opportunity to express their views and cultivate their interests."
• http://www.teachersmonthly.com 2
Head Thoughts - Personal Reflections of a School Headmaster
• blog by Arthur Preston
• "Arthur is currently the headmaster of a growing independent school in Worcester, in the Western
Cape province of South Africa. His approach to primary education is progressive and is leading the
school through an era of new development and change."
• http://headthoughts.co.za/ 3
7.1.2 Maths Blogs
CEO: Circumspect Education Officer 4 - Educating The Future
• blog by Robyn Clark
• "Mathematics teacher and inspirer."
• http://clarkformaths.tumblr.com/ 5
dy/dan - Be less helpful
1 This content is available online at <http://cnx.Org/content/m40358/l.l/>.
2 http:// www.teachersmonthly.com/
3 http://headthoughts.co.za/
4 http://clarkformaths.tumblr.com/
5 http://clarkformaths.tumblr.com/
43
44 CHAPTER 7. MATHS BLOG POSTS
• blog by Dan Meyer
• "I'm Dan Meyer. I taught high school math between 2004 and 2010 and I am currently studying at
Stanford University on a doctoral fellowship. My specific interests include curriculum design (answering
the question, "how we design the ideal learning experience for students?") and teacher education
(answering the questions, "how do teachers learn?" and "how do we retain more teachers?" and "how
do we teach teachers to teach?")."
• http://blog.mrmeyer.com 6
Without Geometry, Life is Pointless - Musings on Math, Education, Teaching, and Research
• blog by Avery
• "I've been teaching some permutation (or is that combination?) of math and science to third through
twelfth graders in private and public schools for 11 years. I'm also pursuing my EdD in education and
will be both teaching and conducting research in my classroom this year."
• http://mathteacherorstudent.blogspot.com/ 7
Overthinking my teaching - The Mathematics I Encounter in Classrooms
• blog by Christopher Danielson
• "I think a lot about my math teaching. Perhaps too much. This is my outlet. I hope you find it
interesting and that you'll let me know how it's going."
• http://christopherdanielson.wordpress.com 8
A Recursive Process - Math Teacher Seeking Patterns
• blog by Dan
• "I am a High School math teacher in upstate NY. I currently teach Geometry, Computer Programming
(Alice and Java), and two half year courses: Applied and Consumer Math. This year brings a new
21st century classroom (still not entirely sure what that entails) and a change over to standards based
grades (#sbg)."
• http://dandersod.wordpress.com 9
Think Thank Thunk — Dealing with the Fear of Being a Boring Teacher
•
blog by Shawn Cornally
"I am Mr. Cornally. I desperately want to be a good teacher. I teach Physics, Calculus, Programming,
Geology, and Bioethics. Warning: I have problem with using colons. I proof read, albeit poorly."
http://101studiostreet.com/wordpress/ 10
6 http://blog. mrmeyer.com/
7 http://mathteacherorstudent. blogspot.com/
8 http://christopherdanielson. wordpress.com/
9 http://dandersod.wordpress.com/
10 http:// 101studiostreet.com/wordpress/
Chapter 8
Solutions 1
8.1 Solutions
The solutions to the exercises can be found embedded throughout the textbooks on Connexions.
The Maths textbook can be accessed at http://cnx.org/content/colll306/latest/ 2
1 This content is available online at <http://cnx.Org/content/m40361/l.l/>.
2 Siyavula textbooks: Grade 10 Maths [CAPS] <http://cnx.org/content/colll306/latest/>
45
46
INDEX
Index of Keywords and Terms
Keywords are listed by the section with that keyword (page numbers are in parentheses). Keywords
do not necessarily appear in the text of the page. They are merely associated with that section. Ex.
apples, § 1.1 (1) Terms are referenced by the page they appear on. Ex. apples, 1
A Assessment, § 2(9)
B Blog posts, § 7(43)
C CAPS, § 8(45)
Citizen cyberscience,
F FullMarks, § 5(33)
4(29)
G Grade 10, § 1(1), § 8(45)
Grade 10 Maths, § 2(9), § 3(25), § 4(29),
§ 5(33), § 6(37), § 7(43)
M Mathematics, § 1(1), § 8(45)
R Rich media, § 6(37)
S Siyavula, § 1(1), § 2(9), § 3(25), § 4(29),
§ 5(33), § 6(37), § 7(43)
Solutions, § 8(45)
T Teachers' Guide, § 1(1), § 2(9), § 4(29), § 5(33)
Teachers' Guides, § 3(25), § 6(37), § 7(43)
ATTRIBUTIONS 47
Attributions
Collection: Mathematics Grade 10 Teachers' Guide - Siyavula WebBooks
Edited by: Bridget Nash
URL: http://cnx.org/content/colll341/l-l/
License: http://creativecommons.Org/licenses/by/3.0/
Module: "TG Maths - Overview"
Used here as: "Overview"
By: Bridget Nash
URL: http://cnx.org/content/m40349/l-l/
Pages: 1-8
Copyright: Bridget Nash
License: http://creativecommons.org/licenses/by/3-0/
Module: "TG Maths - Assessment and Assessment Support"
Used here as: "Assessment and Assessment Support"
By: Bridget Nash
URL: http://cnx.org/content/m40351/l-l/
Pages: 9-24
Copyright: Bridget Nash
License: http://creativecommons.org/licenses/by/3-0/
Module: "TG Maths - Chapter Contexts"
Used here as: "Chapter Contexts"
By: Bridget Nash
URL: http://cnx.org/content/m40352/l-l/
Pages: 25-27
Copyright: Bridget Nash
License: http://creativecommons.org/licenses/by/3-0/
Module: "On the Web, Everyone can be a Scientist"
By: Bridget Nash
URL: http://cnx.org/content/m40353/Ll/
Pages: 29-31
Copyright: Bridget Nash
License: http://creativecommons.org/licenses/by/3-0/
Module: "FullMarks User Guide"
By: Bridget Nash
URL: http://cnx.org/content/m40356/l-l/
Pages: 33-35
Copyright: Bridget Nash
License: http://creativecommons.org/licenses/by/3-0/
Module: "Rich Media"
By: Bridget Nash
URL: http://cnx.org/content/m40357/Ll/
Pages: 37-41
Copyright: Bridget Nash
License: http://creativecommons.org/licenses/by/3-0/
48 ATTRIBUTIONS
Module: "TG - Maths Blog Posts"
Used here as: "Maths Blog Posts"
By: Bridget Nash
URL: http://cnx.org/content/m40358/l-l/
Pages: 43-44
Copyright: Bridget Nash
License: http://creativecommons.Org/licenses/by/3.0/
Module: "TG Maths - Solutions"
Used here as: "Solutions"
By: Bridget Nash
URL: http://cnx.org/content/m40361/l-l/
Page: 45
Copyright: Bridget Nash
License: http://creativecommons.org/licenses/by/3-0/
Mathematics Grade 10 Teachers' Guide - Siyavula WebBooks
This collection is the Teachers' Guide for the Siyavula WebBook - Grade 10 Maths.
About Connexions
Since 1999, Connexions has been pioneering a global system where anyone can create course materials and
make them fully accessible and easily reusable free of charge. We are a Web-based authoring, teaching and
learning environment open to anyone interested in education, including students, teachers, professors and
lifelong learners. We connect ideas and facilitate educational communities.
Connexions's modular, interactive courses are in use worldwide by universities, community colleges, K-12
schools, distance learners, and lifelong learners. Connexions materials are in many languages, including
English, Spanish, Chinese, Japanese, Italian, Vietnamese, French, Portuguese, and Thai. Connexions is part
of an exciting new information distribution system that allows for Print on Demand Books. Connexions
has partnered with innovative on-demand publisher QOOP to accelerate the delivery of printed course
materials and textbooks into classrooms worldwide at lower prices than traditional academic publishers.